Multidisciplinary design optimization: Difference between revisions

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In [[complex analysis]], a '''zero''' (sometimes called a '''root''') of a [[holomorphic function]] ''f'' is a [[complex number]] ''a'' such that ''f''(''a'') = 0.
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==Multiplicity of a zero==
A complex number ''a'' is a '''simple zero''' of ''f'', or a '''zero of multiplicity 1''' of ''f'', if ''f'' can be written as
 
:<math>f(z)=(z-a)g(z)\,</math>
 
where ''g'' is a [[holomorphic function]] ''g'' such that ''g''(''a'') is not zero.
 
Generally, the '''[[multiplicity (mathematics)|multiplicity]]''' of the zero of ''f'' at ''a'' is the positive integer ''n'' for which there is a holomorphic function ''g'' such that
 
:<math>f(z)=(z-a)^ng(z)\  \mbox{and}\ g(a)\neq 0.\,</math>
 
The multiplicity of a zero ''a'' is also known as the '''order of vanishing''' of the function at ''a''.
 
==Existence of zeros==
The [[fundamental theorem of algebra]] says that every nonconstant [[polynomial]] with complex coefficients has at least one zero in the [[complex plane]]. This is in contrast to the situation with [[real number|real]] zeros: some polynomial functions with real coefficients have no real zeros. An example is ''f''(''x'') = ''x''<sup>2</sup> + 1.
 
==Properties==
An important property of the set of zeros of a holomorphic function of one variable (that is not identically zero) is that the zeros are isolated. In other words, for any zero of a holomorphic function there is a small disc around the zero which contains no other zeros.
There are also some theorems in complex analysis which show the connections between the zeros of a holomorphic (or meromorphic) function and other properties of the function. In particular [[Jensen's formula]] and [[Weierstrass factorization theorem]] are results for complex functions which have no counterpart for functions of a real variable.
 
==See also==
* [[Root of a function]]
* [[Pole (complex analysis)]]
* [[Hurwitz's theorem (complex analysis)]]
* [[Rouché's theorem]]
* [[Filter design]]
* [[Nyquist stability criterion]] in [[control theory]]
* [[Marden's theorem]]
* [[Sendov's conjecture]]
* [[Gauss–Lucas theorem]]
 
==References==
*{{cite book |last=Conway |first=John |title=Functions of One Complex Variable I |year=1986 |publisher=Springer |isbn=0-387-90328-3}}
*{{cite book |last=Conway |first=John |title=Functions of One Complex Variable II |year=1995 |publisher=Springer |isbn=0-387-94460-5}}
 
== External links ==
* {{MathWorld | urlname= Root | title= Root}}
* [http://math.fullerton.edu/mathews/c2003/SingularityZeroPoleMod.html Module for Zeros and Poles by John H. Mathews]
* [http://directory.fsf.org/wiki/ZerSol Free C++ solver for zeros of an analytic function (with C and Fortran bindings) by Ivan B. Ivanov]
[[Category:Complex analysis]]
[[Category:Zero]]

Latest revision as of 01:58, 10 September 2014

Are we always having problems with your PC? Are you constantly seeking methods to grow PC performance? Then this really is the post you're lookin for. Here we are going to discuss a few of the most asked concerns whenever it comes to having you PC serve you well; how can I make my computer quicker for free? How to create my computer run quicker?

You usually discover which there are registry products which are free plus those that you will have to pay a nominal sum for. Some registry products offer a bare bones system for free with the option of upgrading to a more advanced, efficient version of the same program.

So what if you look for when you compare registry cleaners. Many of the registry cleaners available now, have rather similar qualities. The main ones that you need to be seeking are these.

In purchase to remove the programs on the computer, Windows Installer should be in a healthy state. If its installation is corrupted you might obtain error 1721 in Windows 7, Vista and XP throughout the program removal procedure. Simply re-registering its component files would resolve a problem.

Another usual cause of PC slow down is a corrupt registry. The registry is a really important component of computers running on Windows platform. When this gets corrupted the PC will slowdown, or worse, not begin at all. Fixing the registry is simple with all the utilize of the program and tuneup utilities.

Another key element whenever you compare registry cleaners is having a center to manage a start-up jobs. This merely means which you can choose what programs you'd like to begin whenever you begin your PC. If you have unwanted programs starting whenever you boot up the PC this usually lead to a slow running computer.

The first reason a computer may be slow is considering it needs more RAM. You'll see this issue right away, incredibly should you have lower than a gig of RAM. Most modern computers come with a least that much. While Microsoft states Windows XP may run on 128 MB, it and Vista want at least a gig to run smoothly and enable you to run multiple programs at when. Fortunately, the cost of RAM has dropped significantly, plus we can get a gig installed for $100 or less.

If you like to have a computer with fast running speed, you'd better install a superior registry cleaner to wash the useless files for you. As long as you take care of the computer, it can keep inside good condition.

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